Classical electrodynamic theory may be regarded as central to physics. An electric field (E) arises from an electric charge density (ρ). Charge motion creates an electrical current density (J) that drives dynamical changes in E and the magnetic field, B. The classical electrodynamic model is based on a coupled set of partial-differential equations for the quantities (B, E, J, ρ).
Classical electromagnetics predicts no wave creation by radial motion of a charged sphere. More specifically, spherical symmetry of the radial electric field on a charged, oscillating sphere implies a curl-free electric field (∇×E=0), which in turn yields no variation in magnetic field from Faraday's law (∇×E=−∂B/∂t=0), corresponding to no magnetic wave. Thus, the Poynting vector, E×B/μ, is zero, resulting in no classical electromagnetic radiation. This statement applies more generally to no creation of electrical waves by radial motion of any extended charge distribution. An electrically equivalent antenna may include a classical linear monopole that is driven by a sinusoidal current to put charge onto and remove charge from the linear conductor (antenna).